Beyond the largest Lyapunov exponent: Entropy-based diagnostics of chaos in Hénon-Heiles and N-body dynamics

The largest Lyapunov exponent is widely used to diagnose chaos in gravitational dynamics; however, in mixed phase spaces and finite-N systems, it does not always provide a complete description of the orbital complexity and phase-space transport. Entropy-based diagnostics could offer a complementary perspective. We investigate whether trajectory-based information entropy can provide a useful diagnostic of chaos in gravitational systems and how it relates to the largest Lyapunov exponent as a function of orbital energy and of the number of degrees of freedom. We computed the largest Lyapunov exponent and a coarse-grained Shannon entropy for ensembles of trajectories in the Hénon-Heiles potential and for test-particle orbits in live N-body realisations of a Plummer model. We compared the dependence of both quantities on the orbital energy as well as on the particle number for the N-body case in particular. In the Hénon-Heiles system, the Shannon entropy follows the transition from weak to widespread chaos and exhibits an energy dependence that closely mirrors that of the largest Lyapunov exponent. For test-particle orbits in live N-body potentials, both diagnostics indicate stronger chaos for more tightly bound trajectories. However, their dependence on N differs: the largest Lyapunov exponent remains nearly constant over the explored range of particle numbers, whereas the Shannon entropy decreases monotonically as N increases. These results show that the Shannon information entropy could serve as a complement to the largest Lyapunov exponent and could also better capture changes in global phase-space mixing, especially in systems where the leading Lyapunov exponent alone is not sufficiently informative. It provides a promising alternative for diagnosing chaos when tangent-space dynamics is unavailable, prohibitively expensive, or slowly convergent. In particular, because this entropy measure relies only on time series of phase-space coordinates, it is naturally suited to systems with densely sampled trajectories, such as minor bodies in the Solar System.